1The idea in one paragraph

A heavy nucleus such as uranium-235 is less tightly bound, per nucleon, than a middle-sized one. Split it in two and the nucleons end up more tightly bound, so the products have less mass than what went in. The missing mass is released as energy, mostly as the kinetic energy of the two fragments. That is nuclear fission. Each fission also releases two or three neutrons, which can split further nuclei: a chain reaction. A nuclear power plant holds that chain at a steady rate with control rods, slows the neutrons with a moderator, carries the energy to a heat exchanger that raises steam for a turbine, and keeps the radiation in with shielding. The fragments left behind are radioactive, some for a very long time.

2Where the energy is stored

The energy of fission is stored in the way the nucleons are arranged: it is nuclear potential energy, set by how strongly the strong nuclear force holds each nucleon against the electric repulsion between protons. Binding energy measures it.

The binding energy of a nucleus is the energy needed to separate it completely into its protons and neutrons, which is also the energy released when they come together. Figure 1 plots binding energy per nucleon against nucleon number A.

Uranium-235 sits at about 7.6 MeV per nucleon; krypton-92 and barium-141 sit at about 8.3 to 8.5 MeV per nucleon. Split uranium into two middle-sized nuclei and every nucleon moves up the curve by roughly 0.8 MeV.

Why is uranium less tightly bound? The strong force has a very short range, so each nucleon is held only by its nearest neighbours, and that attraction per nucleon is about the same in any nucleus that is not tiny. Electric repulsion acts between every pair of protons across the whole nucleus. With 92 protons it adds up and loosens every nucleon. After the split, the two pieces no longer repel each other from inside one nucleus.

Energy is released when nucleons end up more tightly bound. Energy released = total binding energy after − total binding energy before, and it appears as a loss of mass.

Keep the sign straight: binding energy is energy you would have to supply to take a nucleus apart, so more binding means a larger mass defect and less mass.

3Spontaneous and neutron-induced fission

Spontaneous fission is a heavy nucleus splitting on its own. It is random, like any radioactive decay. Uranium-238 does it, but very rarely. Californium-252 splits spontaneously in a few per cent of its decays, which makes it useful as a small neutron source for starting a reactor.

Neutron-induced fission is what a reactor runs on. Figure 2 follows one event.

1. A slow neutron is absorbed by uranium-235. It has no charge, so there is no repulsion to overcome and even a very slow neutron gets in.
2. The result is uranium-236 in an excited state, ²³⁶U*.
3. The excited nucleus oscillates, stretches, and the electric repulsion between its two ends pushes them apart. It splits into two unequal fragments plus two or three neutrons.

One possible outcome:

Check it balances. Nucleon numbers: 1 + 235 = 236, and 141 + 92 + 3 = 236. Proton numbers: 92 = 56 + 36.

Uranium-235 splits in dozens of ways, usually into one fragment with A near 90 to 100 and another near 135 to 145, with 2.4 neutrons on average. Each fission releases about 200 MeV, over 80% of it as kinetic energy of the fragments. Collisions with the surrounding atoms quickly turn that kinetic energy into internal energy of the fuel.

4Calculating the energy released

Two routes give the same answer.

Route 1: from masses. Energy released = (mass before − mass after) × c², or × 931.5 MeV per u.

Worked example 1. Use the reaction above. Masses: uranium-235 = 235.043930 u, barium-141 = 140.914411 u, krypton-92 = 91.926156 u, neutron = 1.008665 u.

Keep every decimal place until the subtraction, because the answer is a small difference between two large numbers. Count the neutrons: three out means three neutron masses on the right. These are atomic masses, which include electrons, but the 92 electrons of uranium match the 56 + 36 of barium and krypton and cancel.

Route 2: from binding energies. Energy released = total binding energy after − total binding energy before. Total binding energy = binding energy per nucleon × A. A free neutron has none.

Worked example 2. Binding energy per nucleon: uranium-235 7.59 MeV, barium-141 8.33 MeV, krypton-92 8.51 MeV.

The routes agree to within rounding. A negative answer means you subtracted the wrong way round.

Worked example 3: energy from a kilogram. If every nucleus in 1.0 kg of uranium-235 undergoes fission at 200 MeV each:

Burning a kilogram of coal releases a few times 10⁷ J: fission gives millions of times more per kilogram.

5Chain reactions

Each fission is started by one neutron and releases two or three. If at least one of them causes another fission, the process sustains itself: a chain reaction. Figure 3 shows the two ways it can run.

Not every neutron causes fission: some escape through the surface of the fuel, some are absorbed by uranium-238 or other materials. What matters is k, the average number from each fission that go on to cause another. Figure 4 shows three cases.

• k > 1, supercritical: the rate grows every generation, and a generation lasts a tiny fraction of a second. Uncontrolled, this is a nuclear weapon.
• k = 1, critical: each fission leads to exactly one more. The power is constant. This is a reactor in steady operation.
• k < 1, subcritical: the chain dies away.

Neutrons are made throughout the fuel but escape through its surface, and a bigger lump has less surface per unit volume. So there is a critical mass: the smallest mass of fissile material, for a given shape and density, that can sustain a chain reaction. Natural uranium is only 0.7% uranium-235, the rest uranium-238, which absorbs many neutrons without splitting, so reactor fuel is enriched to a few per cent uranium-235.

6Inside a nuclear power plant

A reactor must make fission likely, hold k at exactly 1, get the energy out, and keep the radiation in. The guide names the part that does each; Figure 6 puts them in place.

The moderator slows the neutrons. Fission neutrons are fast, about 2 MeV. Uranium-235 is far more likely to absorb a slow thermal neutron, about 0.025 eV, similar to the kinetic energy of gas particles at room temperature. The neutrons lose energy by colliding many times with the nuclei of the moderator: water, heavy water or graphite.

The moderator must be made of light nuclei. In an elastic collision, a small ball hitting a much heavier one bounces off with almost all its speed; hitting an equal mass head-on, it stops and hands over all its kinetic energy. For a head-on elastic collision with a nucleus of nucleon number A, conservation of momentum and kinetic energy give the fraction the neutron keeps, ((A − 1) ÷ (A + 1))². Figure 5 shows it for four targets.

Worked example 4. A 2.0 MeV neutron collides head-on, every time, with carbon-12 nuclei. How many collisions bring it to 0.025 eV?

Real collisions are mostly glancing, so about twice as many are needed. Uranium, keeping 98% each time, would need thousands.

The control rods hold k at 1. They are made of a strong neutron absorber, such as boron or cadmium, and moved in and out of the core. Further in, they absorb more neutrons, k falls and the power drops; further out, the power rises. In an emergency they drop fully in and the chain reaction stops. Control rods remove neutrons; the moderator slows them. Swapping the two is the commonest error in this subtopic.

The coolant and heat exchanger get the energy out. A coolant is pumped through the core and carries away the energy that heats the fuel rods. In Figure 6 the coolant is water, kept at high pressure so it stays liquid at around 300 °C, and it is also the moderator. In the heat exchanger the hot coolant flows through pipes surrounded by a separate supply of water; energy passes through the pipe walls and that second supply boils. Its steam drives the turbine and generator, is condensed by cooling water, and returns. The two loops never mix, so the radioactive coolant never reaches the turbine.

The shielding keeps the radiation in. The core emits intense gamma radiation and neutrons. A thick steel pressure vessel inside thick concrete absorbs them, keeping the dose to workers and the public very low, and the containment building is designed to hold radioactive material in if something goes wrong.

Figure 7 traces the energy: binding energy, then kinetic energy of fragments, internal energy of fuel and coolant, steam, the turbine's kinetic energy, and finally electrical energy. The efficiency is about a third, because, as in any steam power station, much of the energy goes to the cooling water to condense the steam.

Worked example 5: from power to fuel. A reactor produces 3.2 GW of thermal power and 1.1 GW of electrical power, with 200 MeV per fission. Determine (a) the efficiency, (b) the fissions per second, (c) the mass of uranium-235 used per day.

That answers the guide's linking question on the rate of energy production: energy per fission, from binding energy, divided into the power gives the rate.

7The products of fission and their management

Why the products are radioactive. Heavy nuclei need more neutrons than protons to be stable, because extra neutrons add strong-force attraction without electric repulsion. Uranium-235 has 143 neutrons to 92 protons, a ratio of about 1.55; stable middle-sized nuclei have about 1.3 to 1.4. The fragments inherit roughly uranium's ratio, so they have too many neutrons. Figure 8 shows them above the line of stability.

A neutron-rich nucleus becomes stable by beta-minus decay: a neutron turns into a proton, emitting an electron and an antineutrino, one step down and to the right in Figure 8. Barium-141 takes three steps to stable praseodymium-141. Most of these decays also emit gamma rays. So fission products:

• are beta-minus and gamma emitters, because they are neutron-rich;
• have half-lives from fractions of a second to millions of years: iodine-131 about 8 days, caesium-137 and strontium-90 about 30 years, technetium-99 about 200 000 years;
• have very high activity when fresh, so they keep producing decay heat after the chain reaction stops. In 2011 at Fukushima Daiichi in Japan the reactors shut down when the earthquake struck, but the tsunami cut the power to the cooling pumps, and decay heat alone melted fuel in three reactors;
• come mixed with plutonium-239, made when uranium-238 absorbs a neutron and then undergoes two beta decays. It is an alpha emitter with a half-life of about 24 000 years, it is fissile, and it could be used in a weapon.

How the waste is managed. Low-level waste (clothing, tools) is compacted and buried near the surface. Intermediate-level waste (reactor parts, filter resins) is sealed in cement. High-level waste, the spent fuel, follows Figure 9.

It spends years in a cooling pond, where water removes the decay heat and absorbs the radiation; then decades in air-cooled steel and concrete dry casks. Some countries then reprocess it, separating uranium and plutonium for reuse and melting the fission products into glass (vitrification); others seal the whole fuel in canisters. The long-term plan is a deep geological repository, hundreds of metres down in stable rock. Finland has built the first, at Onkalo.

The impact of long-term storage. Weigh both sides.

• Some waste stays hazardous for tens of thousands of years, longer than any human institution has lasted. A repository must stay sealed through changes in climate and groundwater that nobody can test in advance.
• It passes a burden, and a cost, to generations who did not use the electricity: an ethical question, not only a technical one.
• Plutonium must be guarded against theft.
• Against that: the volume is small for the energy produced, it is solid, contained and tracked, whereas the waste of fossil fuels, carbon dioxide, goes straight into the atmosphere. And its activity only ever decreases.

8Fission and climate change

The guide asks to what extent fission has a role in addressing climate change. For: a fission station emits almost no carbon dioxide while running, and its life-cycle emissions per unit of electricity are similar to wind and far below coal or gas; its fuel is extremely energy-dense; it generates steadily in any weather, complementing wind and solar. Against: long-lived waste, the small but serious risk of accidents, the link to weapons, long and expensive construction, and uranium being a finite, mined resource. A balanced conclusion names the trade-off: large amounts of low-carbon electricity, paid for with a waste and safety problem that outlasts the station by millennia.